Is the word vestir the same in Brazilian and European Portuguese?

No.

The primary differences are grammatical and phonetic.

Grammar: The main difference lies in clitic placement (the position of object pronouns). In Brazilian Portuguese, it is standard to use proclisis, where the pronoun comes before the verb (e.g., "me vesti"). In European Portuguese, enclisis is the standard, where the pronoun follows the verb (e.g., "vesti-me").

Usage: While "vestir" is used in both dialects, the typical vocabulary for related concepts differs. A person from Portugal is much more likely to use the verb usar (to wear) when talking about habitual clothing or servir (to fit) when discussing if a garment is the right size. Brazilians frequently use "vestir" or "colocar" (to put on) for both the action and the state of wearing.

Pronunciation: Brazilian Portuguese generally features more open vowels and a more "musical" rhythm, whereas European Portuguese features more "reduced" or "swallowed" unstressed vowels, making the speech sound more consonant-heavy.

Brazilian Portuguese Examples:

1. Eu me vesti muito rápido. (I dressed very quickly.)
2. Ela sempre se veste de preto. (She always dresses in black.)
3. Você se vestiu bem para o jantar. (You dressed well for the dinner.)
4. Ele se veste de uma forma muito casual. (He dresses in a very casual way.)
5. Essa calça não me veste bem. (These pants don't fit me well.)

European Portuguese Examples:

1. Eu vesti-me muito rápido. (I dressed very quickly.)
2. Ela usa sempre preto. (She always wears black.)
3. Vestiste-te bem para o jantar. (You dressed well for the dinner.)
4. Ele veste-se de uma forma muito casual. (He dresses in a very casual way.)
5. Esta calça não me serve bem. (These pants don't fit me well.)